
Subject: cube roots
Hi. My name is Heather Jones. I'm in 11 grade. What do the graphs of cube roots look like?
Hi Heather,
There are (at least) a couple of ways to 'see' this.

Visual. Draw a graph of what y = x^{3} (the cube) looks
like. Now YOU want the graph where x and y switch roles:
y^{3} = x or y = x^{1/3}. (This, by the way, is another standard
notation for cube root. It is the notation you will need if you
want to calculate cube roots of specific numbers on a calculator.)
You can get this 'reversed' graph by taking the first one
(drawn with the same scale on the x and y axes) and putting
down a mirror on y = x. Now reflect the first graph in this mirror.
Y becomes x, x becomes y and the first graph y=x^{3} becomes
y^{3}=x.
Another way to see this is to draw the graph of y = x^{3} on a transparency that you would use in an overhead projector. Now switch the roles of x and y by labeling the horizontal axis y, and the vertical axis x. Finally turn the transparency over and rotate it so that the axis labeled y is vertical, with positive upwards, and the axis labeled x is horizontal with positive to the right. What you see is the graph of y = x^{1/3}.
Harley
This is a general technique for turning the graph of ANY function
into the graph of the 'inverse' function. Works for y=x^{2} to y^{2}=x
(or equivalently y=x^{1/2}  the square root).
Just as the graph of y=x^{3} goes up far faster than y=x^{2} or y=x,
the graph of y=x^{1/3} grows SLOWER than the graph of y=x^{1/2}
which, in turn, is slower than y=x.

Numerical. Take pairs of points from y=x^{3}.
(2,8), (1,1), (0,0), (1,1), (2,8) etc.
Reverse each of the pairs:
(8,2), (1,1) (0,0) (1,1) (8,2) etc.
These reversed points (the mirror images in 1.) are points on the
graph: cube root of 8 is 2, etc.
Use these points (and others as necessary) to plot the graph.
(3) With a calculator, you can get points on the graph just
by plugging in points, and knowing that cube root of x
is the same as x to the exponent (1/3). Again, enough points
will give you information for a sketch of the graph.
Cheers,
Walter
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